As Mariano was mentioning what you're referring to is a special case of twisted derivations. Since you seem to be interested in differential calculus than a possible reasonable appearance of twisted derivations in quantum group theory emerges if one consider twisted Hochschild homology and its cyclic version as well (e.g. http://openscholarship.wustl.edu/math_facpubs/4/ for some further explanations on relations).

Twisted Hochschild and cyclic homology may play the role of substitute of differential forms and De Rham complex, and were computed in a bunch of examples (again just one citation to give a starting point http://www.maths.gla.ac.uk/~ukraehmer/twisted2final.pdf).

One of the most interesting features of this twisted homology is that while usual Hochschild homology of quantum groups and quantum spaces faces a "dimension drop" problem, i.e. the homological dimension of quantum spaces is lower than the classical ones, the twisted one does not have this problem, in computed case, and has the same dimension as the classical one.

A possible explanation of this fact lies in the connection between quantum algebr and Poisson geometry: the Poisson manifolds underlying quantum groups and spaces are, in general, not unimodular, so that Poisson homology and cohomology fail to be one dual to the other.

A part form the Poisson geometrical interpretation, the dimension issue itself, together with its relation with the "modular aspects" of the theory (at an operator algebraic level quantization of the Poisson modular class is the Tomita-Takesaki operator) induced some people to work on new definition of spectral triples relying on twisted Hochschild (cyclic) theory rather than the usual one. They are named twisted spectral triples and I guess you can easily find a lot of papers dealing with them in many specific examples of quantum groups.

Sorry of all this is somewhat generic; putting details would mean writing a review paper of a whole area of research, rapidly developing in last 10 years...